DEUTSCHES ELEKTRONEN-SYNCHROTRON DE SY DESY 77/7 May 1977 p Recurrences in J/~ + 3~ by II. Z. z., Aydin. Abstract:.. We study the recurre~~~~ decay of the J/~ and show that for this decay gives the following ratios r(j/~ + g~) = 6 x 10- r ~ + p~) r(j/~ + i~) = 1 x 10-4 "'r-;-( J~/,-;~-+-p-~-;-) NOTKESTRASSE 85
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- I - In a recent Letter, Cohen-Tannoudji et al. l) have pointed out that the hadronic decays of the J/~ may provide us with the possibility of studying heavy resonances such as recurrences of the r~w, B etc. It is obvious that since the J/'f' is heavy, enough energy is available in a given subchannel of the l/\v decay in order to reach the higher recurrences of a Regge trajectory, The more important part of the argument is that since the c quark in the J/~ cannot end up in the final state hadrons (Fig. 1), the exchanges between the final state hadrons have no twist and therefore the subchannels are maximally resonating. The experimental data on the 3n. decay of the J/'4', showing that 70% of the 3n: decay is the yn mode gives credence to this argument. (Another support to this argument comes also ), from the decay -i' 31(. Very recently the Orsay group J) has reached the conclusion that greater than 80% of the 3'}"( decay of the 4 proceed through the pn mode). In this short note we want to study the problem quantitatively for the recurrences f(rj ~ ~t3-), A.lS-),... appearing in the 31\ decay of the J/yp. The preliminary analysis of an experiment at DESY 4 ) gives an upper limit, i.e., I( J/'l'-'> ~ T<) T' (J/'i' ~ J'") (I) Using the widlh formulas) for the decay of a spin-} meson into a spin-1 and a spin-0 meson, we can write r(j/'l'~\jt<) \' (Jf'\' _., j'n) Gnn < o.s- -~,_- G'I'fT\ 4 15 1 P.._ R' f ( :,:-) () " f Recurrences in J I"\',. 3 n z. z. Aydin + II. Institut fur Theoretische Physik der Universitat Hamburg Abstract: We study the recurrences _p(t-), 'J-(3-),.\..(5-):-etc. in the "3n. decay of the J/l}J and show that the Virasoro dual amplitude for this decay gives the following ratios r ( J/'\1--+ 'J 1\) r(j/'1'-- f"') 6 X 10 -. r ( J/4' _, i- 1\ ; I( J/<\' _, y " ) i X io -4 +)Alexander von Humboldt Fellow. On leave from Ankara University and High Energy Physics Group of the Scientific and Technical Research Council of Turkey.
.. f - - ~"'f/'rf 1 ) leads to have poles in all three channels which correspond to the (s,t), (s,u) and is given by t'"'"p E P f> p V(,,t,u) - 3 - f(~-~<x(s~ \'(~-~cx(t0 f(7-7 x(c.)) \, (-\) \("::;-'>. the Virasoro amplitude is regular if o{(s) + o((t) +Q(lu.);::: N--::. odd integer number. Since o(}'(s)-=. 0.47+ 0.88 s, N is not at all an odd present. In the center of mass frame of the s-channel, Eq. (3) can be written as where q and q' are the magnitudes of the center of mass three- momenta before and after the collision and e is the center of mass scattering \-0\(s) \)l,,t,u) 0 - (... \ (,. \ (4) 1(1-Jo((S)-10\(01 ~-~o((s)-to((u~ r\1-iu<(t)-~o<(uy Here we have only the f -trajectory in all three channels. We now expand the amplitude (4) as a sum of the pole terms: ~ " Vcs,t,u.) = ~ L ( Ht)-.x(s) fh-l o<\t)+.r-1] i l c((u)tx-1.1 -n-::.0 1( 1 H «ltl +o(lu) 1) (5) One should note that because of the gamma function in the denominator integer (actually N~IO) foe Jj\ _,. 31\, so that poles are really A = JS qc(.,;..,e (vi +V 3 +V 5 ) (6) angle; and v 1,v 3,v 5,... stand for the pole terms at o((s) = 1,3,5,... (in Eq. 5) with residues evaluated also in the center of mass frame. Now if we expand our helicity amplitude A into partial waves as: A L_ "' ~ r,, t l (e) 10 (7) "'e can easily identify the so-called relative strengths of the where ~ ( ~) is the magnitude of the 3-momentum of the g (~)in the rest frame of the J/+, It seems that the factor to a large enhancement. However, v1e have to know also (.. " :j > the factor G.., /&.,~1\': 'tyn' It is known that there is some arbitrariness in the definition of coupling constants because of their dimensions; namely one can define them involving arbitrary masses. Therefore, it would be nice if one extracts these coupling constants from a dynamical theory. This can be achieved in a dual model, since a dual model relates dynamically the particles with different spins, As pointed out in Ref. (1), the Virasoro amplitude, and not the Veneziano amplitude, should be used for the J{'t'...l) '311:. The reason for this is that the process J{lf4?,n: has very different duality properties in comparison to the process W 4 1n. For example, in the w ~ '3n decay one needs three diagrams in the lowest order in order (u,t) terms of the Veneziano amplitude, but in the J/IJ( ~ ~'lt: case one diagram has poles in all channels. Furthermore, since o((':!)+«(t)to<.lu)_,t, for J/'-t' ~ 31T, it is not possible to impose the necessary condition to cancel all the undesired poles at even integer values of d.. in the case of the Veneziano amplitude. The Virasoro amplitude however, does not suffer from the defect of having poles at even integers. The amplitude foe the reaction J/'\'(p) +TI(-p 1 )_,. T<(!' ) +T<(p 3 ) A (s,t,u) _ I' " o- f _, ) (3) where s = (p 1 +p ), t = (p +p 3 ), u = (p 1 +p 3 ), and the scalar amplitude V(s,t,u) has the Virasoro form:
- 5 - To be more precise,...,-e evaluate the Born terms for the p, g and i poles, and Bov"' Bu1 n Bo~ n equate the r r and r so found to the \1 ' 31 S? corresponding quantities in Eq. (8). We arrive in this way at the following relations: G q,., -== 'Pfrr f \(a;') i 0(, (io) G'~'~" G~"" == f3 4 r( 5t) G lf lt~ G. "" f3 'X '4 16 r('f-:) - 4 - resonances (or partial wave residues). r " " ' ' :~. ~1 -::: r my 'j~ 91 f(\~,nj,,> 'I> \3 -::: ~ '"'} 9~ 9~ 5{3 1( 5 -~) d..,z (8) r55 -== I' "" 9, 1<.1 i5 5,5 16 If o<. --,4 r( '~~) where o1..' is the slope of the f -trajectory. At this point we can say that roughly ~ 1 ~ T;l : \ 55 are the ratios for the densities of.. l. h Jj po1nts around s = mf', m:} and mi 1n the Da Hz plot of t e 't'~ 3n decay. Numerically we have 1. - - - "' 9 "~a _,33 ""400 1' 33 1 ss (9) These imply that there is no hope to see the higher recurrences beyond the g(j-) in the Dalitz plot.
where "'';}" G decay widths or, for, ~.--."35. G'3''" - n - \(~""""J using the elastic widths 6 ) obtained from the Veneziano amplitude ~ Gf"" G,_ ~nn \! "'' } ( 13) For N 10, Eq. (ll) gives, with the value of Eq. (1) or Eq. (13), G'f~rr "" 0.035 I( J/'f--> if 1T) T' (J/'1'-> 5'") which is one order of magnitude smaller than the e:-.-perimental upper limit (1). ( 14) (I 5) Similarly ~,e find f(j/<t'->~n) G.P'"' Gtnn - 7 - = 3\S4 o<.'4 In conclusion, we emphasize that the Virasoro model for the 3n: decay opportunity of studying only the ru-) and i (5-) and the higher recurrences of f(1-). Acknowledgement discussions and reading the manuscript. I also thank A.Ali for reading -4 (I 7) 1 ' 10 (16) In order to get the last result we used the appropriate formula corresponding to Eq. (), Eq. (10) and the ratio of the J/'t' suggests that this decay can provide us with the ';j (,-), but not the I would like to thank G. Kramer for suggesting the problem, the manuscript and J.G. KOrner for a discussion. I (J!'!', fh ) Thus we obtain the ratio G.. - ~ l 1(3;') ~ Gf"" (I I) 16 1(5-N) Gl 'l'pn. ';}'"' GrH I G~"" can be found either using the known experimental.,1 Gf"" - % m~ 9'} ~- ~- f(? -->Tin) <=:: H.5 G \1 4 ""'} 9f ( 1) tl'l\ ~ 111\. In the latter case G'~' J'" Finally, if we use this value in Eq. () we obtain 0. () 6
J/IP ~,c Tt Tt ~c Tt Fig.1-8 - References ], G. Cohen-Tannoudji, G. Girardi, F. Hayot, U. Maor and A. Morel, Phys. Lett, 6B, 343 (1976).. B. II. Wiik, DESY 76/5; Invited talk at the 18th Int. Conf. on High Energy Physics, Tbilisi, USSR, 1976. 3. S. Jullian, Invited talk at the 18th Int. Conf. on High Energy Physics, Tbilisi, USSR, 1976, 4. W. Bartel, P. Duinker, J. Olsson, P. Steffen, J. Heintze, G. Heinzelmann, R.D. Heuer, R. Mundhenke, H.Reiseberg, B. Schilrlein, A. Wagner and A.H. Walenta, Phys. Lett. 64B, 483 (1976),and private communication. S. G. Carbone, Nuovo Cimento 58A, 3 (1968); T.J. Weare, Nuovo Cimento 56A, 64 ( 1968). 6. K. Igi, Phys. Rev.~. 865 (1977);R.L. Thews, Rutherford Lab. preprint RL-77-006/A (1977).